[论文解读] Restoring similarity in randomized Krylov methods with applications to eigenvalue problems and matrix functions

Laura Grigori, Daniel Kressner+2 more|arXiv (Cornell University)|Jan 15, 2026

Matrix Theory and Algorithms被引用 0

一句话总结

本文提出一种对随机 Arnoldi 过程的相似性恢复修正，使其 Hessenberg 矩阵与标准 Arnoldi 矩阵在相似性意义上等价，并将其应用于特征值问题和矩阵函数计算。

ABSTRACT

The randomized Arnoldi process has been used in large-scale scientific computing because it produces a well-conditioned basis for the Krylov subspace more quickly than the standard Arnoldi process. However, the resulting Hessenberg matrix is generally not similar to the one produced by the standard Arnoldi process, which can lead to delays or spike-like irregularities in convergence. In this paper, we introduce a modification of the randomized Arnoldi process that restores similarity with the Hessenberg matrix generated by the standard Arnoldi process. This is accomplished by enforcing orthogonality between the last Arnoldi vector and the previously generated subspace, which requires solving only one additional least-squares problem. When applied to eigenvalue problems and matrix function evaluations, the modified randomized Arnoldi process produces approximations that are identical to those obtained with the standard Arnoldi process. Numerical experiments demonstrate that our approach is as fast as the randomized Arnoldi process and as robust as the standard Arnoldi process.

研究动机与目标

• Motivate and address the mismatch in similarity between Hessenberg matrices from randomized vs. standard Arnoldi.
• Develop a procedure to restore similarity by enforcing orthogonality of the last Arnoldi vector via a least-squares correction.
• Show equivalence to standard Krylov methods in exact arithmetic for eigenvalue problems and matrix function evaluations.
• Provide backward stability insights for eigenvalue computations within the Krylov–Schur framework.

提出的方法

• Introduce a similarity-restoring correction to the randomized Arnoldi process that enforces U_m^H û_{m+1}=0 by solving a least-squares problem.
• Formulate the corrected decomposition AU_m = U_m Ĥ_m + û_{m+1} c_m^H, with Ĥ_m = H_m + ŷ_m c_m^H.
• Prove that the resulting Ĥ_m is similar to the standard G_m, yielding identical Ritz values in exact arithmetic.
• Apply the corrected method to randomized Krylov–Schur for eigenvalues and to matrix-function evaluations via f(A)b.
• Discuss computational costs, highlighting that the dominant cost is forming U_m^H U_m and that the correction is practical.
• Provide a practical implementation outline including optional sketched reorthogonalization.

Figure 3 : Execution time (s) for KS (left), SRR-KS (middle) and RKS (right) on Hermitian (top) and non-Hermitian (bottom) problems. The X-label and Y-label are $\ell$ and $m$ , the dimension of Krylov subspaces before and after restarting, respectively. The stopping criterion is when the residual n

实验结果

研究问题

• RQ1 Can a similarity-restoring correction make the randomized Arnoldi Hessenberg matrix similar to the standard Arnoldi Hessenberg matrix?
• RQ2 Does the corrected randomized Krylov–Schur method produce Ritz values equivalent to the standard Krylov–Schur in exact arithmetic?
• RQ3 What is the backward stability behavior of the similarity-restored approach in eigenvalue computations?
• RQ4 How does the corrected method impact the computation of matrix functions via Krylov subspaces compared to standard approaches?

主要发现

• The similarity-restoring correction yields a Krylov decomposition that is similar to the standard one, aligning Ritz values with those from standard methods in exact arithmetic.
• The proposed approach preserves the speed advantages of randomized Gram–Schmidt while achieving the robustness and eigenvalue accuracy of the standard Arnoldi process.
• In eigenvalue problems, the corrected randomized Krylov–Schur method matches the convergence behavior of the standard Krylov–Schur in exact arithmetic.
• The residuals and numerical ranges are better controlled due to similarity, mitigating issues such as spurious complex Ritz values when A is Hermitian.
• Numerical experiments indicate the method is as fast as randomized Arnoldi and as robust as standard Arnoldi for eigenvalue problems and matrix-function evaluations.

Figure 4 : Convergence history of residual for KS, SRR-KS and RKS on Hermitian (left) and non-Hermitian (right) problems with $\ell=20$ and $m=30$ . The history of KS and SRR-KS are overlapped.